In Radio Frequency Identification (RFID) applications, it is beneficial to know the three-dimensional space in which a tag will operate with respect to the interrogating transmitter. A meaningful evaluation becomes a very complex problem containing numerous variables including transmitted power, antenna gains, orientation, etc., and many discrete points even for a relatively coarse discretization. One well-known equation used to approximate the power that a tag can receive from an interrogating transmitter is the Friis Equation. However, the commonly used form of the Friis Equation contains assumptions that limit the validity to a single point, orientation, and polarization in space, which is usually the most favorable. These simplifications eliminate the reflection coefficients and polarization terms, and the gains lose their angular dependences.
It has been shown that the power at a receiving antenna, specifically an RFID tag, can be found using the Friis Equation shown below.
                              P          R                =                              P            T                    ⁢                                                                      G                  T                                ⁡                                  (                                                            θ                      T                                        ,                                          ϕ                      T                                                        )                                            ⁢                                                G                  R                                ⁡                                  (                                                            θ                      R                                        ,                                          ϕ                      R                                                        )                                            ⁢                              λ                2                                                                    (                                  4                  ⁢                                                                          ⁢                  π                  ⁢                                                                          ⁢                  r                                )                            2                                ⁢                      (                          1              -                                                                                      Γ                    T                                                                    2                                      )                    ⁢                      (                          1              -                                                                                      Γ                    R                                                                    2                                      )                    ⁢                                                                                                        p                    ^                                    T                                ·                                                      p                    ^                                    R                                                                    2                                              (        1        )            Where:                PR—received power        PT—transmit power        GR(θR,φR)—angular dependent receiver (tag) gain        GT(θT,φT)—angular dependent transmitter gain        ΓT—transmitter reflection coefficent        ΓR—receiver reflection coefficent        {circumflex over (p)}T—transmitter polarization vector        {circumflex over (p)}R—receiver polarization vector        r—distance between the transmitter and receiver        λ—wavelengthGenerally, in RFID applications, the maximum read range is calculated and specified by taking the maximum values of the gain terms and eliminating reflection and polarization loss terms as shown in the following equation.        
                              r          max                =                              λ                          4              ⁢                                                          ⁢              π                                ⁢                                                                      P                  T                                ⁢                                  G                  T                                ⁢                                  G                  TAG                                                            P                min                                                                        (        2        )            
The maximum read range is very commonly used as a figure of merit when comparing the operations of different RFID tags. The main problem with this comparison is that it does not give information on how the tag will operate in general when not optimally located in the main beam with matched polarization. It would be much more beneficial to compare the three-dimensional volume (enclosed by a surface area) of operation for RFID tags, specifically, where the tag can be powered for a given percentage of all possible orientations and polarizations (in discretized form). This can be accomplished by solving the Friis Equation shown in (1). However, the computational problem quickly grows to a size which makes it impractical to solve. As an example, for a tag at a given point in space there are 64,800 gain values for a one-degree resolution (180°×360°). This means the tag can be positioned in one of 64,800 possible orientations with respect to the interrogating transmitter. The problem is complicated further when the effects of polarization are included. At a given point in space with a given gain value (1 of 64,800), there are 360 possible polarization positions for a one-degree resolution. This number can be reduced to 90 possible polarizations by the realization that all four quadrants will produce the same results. Therefore, at a given point in space there are 5,832,000 possible gain and polarization positions. If the problem is then solved for one cubic meter in front of the interrogating transmitter in 10 cm steps, there are 1000 points with each having 5,832,000 possible gain and polarization positions. The result is 5,832,000,000 positions for a one cubic meter volume. If it is assumed that the results are stored in 32-bit (4-byte) floating point numbers, the amount of memory needed would be 23.328 giga-bytes (GB), which is not practical on most computer systems. Given this result, it is obvious that a method is needed for determining the RFID system operating envelope (volume) that is manageable and yet useful for evaluating the performance of the overall system.